Exercises: Evaluating limits

Given the graph of the function $f(x)$ shown below, visually identify the value of the following limits:

1. a) $\lim_{x \to 2^-} f(x)$

1. b) $\lim_{x \to 2^+} f(x)$

1. c) $\lim_{x \to +\infty} f(x)$

Calculate the limit at infinity of the following polynomial function, which represents the evolution of the profits of a tech startup (in thousands of euros) over time $x$:

$$\lim_{x \to +\infty} ( -2x^3 + 50x^2 - 100 )$$

State which is the dominant term and justify why the sign of the leading coefficient determines the end behaviour of the function.

A video game server manages the number of connected players using the function $P(x) = \frac{6x^2 + 20}{2x^2 + 5}$, where $x$ is the time in hours since the launch of an event. Calculate the limit of players as time tends to infinity:

$$\lim_{x \to +\infty} \frac{6x^2 + 20}{2x^2 + 5}$$

In a chemistry laboratory, the concentration of a reactant in a mixture (in mg/l) follows the function $C(t) = \frac{5t + 8}{t^2 + 2}$, where $t$ is the time in minutes. Determine the long-term value to which the concentration tends by calculating:

$$\lim_{t \to +\infty} \frac{5t + 8}{t^2 + 2}$$

Calculate the one-sided limits of the function $f(x) = \frac{3}{x - 5}$ at the point where the denominator vanishes. To do so, study the behavior of $f(x)$ as you approach from the left and from the right, paying attention to the sign of the numerator and the denominator:

1. a) $\lim_{x \to 5^-} \frac{3}{x - 5}$

1. b) $\lim_{x \to 5^+} \frac{3}{x - 5}$

The diagram shows the axes, the vertical asymptote $x=5$ and the direction in which we approach it from each side:

Resolve the indeterminate form $\frac{0}{0}$ by simplifying the fraction through factorising the polynomials (you can use algebraic identities or Ruffini's rule):

$$\lim_{x \to 2} \frac{x^2 - 4}{x^2 - 2x}$$

Based on comparing orders of infinity, determine the value of the following limits without performing complex operations:

1. a) $\lim_{x \to +\infty} \frac{e^x}{x^{10}}$

1. b) $\lim_{x \to +\infty} \frac{\ln(x)}{x^2}$

1. c) $\lim_{x \to +\infty} \frac{2^x}{3^x}$

The altitude of a reconnaissance drone (in metres) with respect to the horizontal distance $x$ covered is given by the function $h(x) = \frac{x^2 - 9}{x - 3}$. Although the drone cannot physically be at position $x = 3$ due to a software restriction, calculate the limit of the altitude as it approaches that point:

$$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$

Mateo is solving the following limit on the board, which presents an indeterminate form of type $1^\infty$:

$$\lim_{x \to +\infty} \left( \frac{2x + 5}{2x + 1} \right)^{3x}$$

To solve it, he carries out the following procedure:

1. He checks that $\lim_{x \to +\infty} \frac{2x + 5}{2x + 1} = 1$ and that the exponent tends to $+\infty$.

2. He applies the auxiliary exponent formula:

$$L = \lim_{x \to +\infty} 3x \cdot \left( \frac{2x + 5}{2x + 1} - 1 \right) = \lim_{x \to +\infty} 3x \cdot \left( \frac{4}{2x + 1} \right) = \lim_{x \to +\infty} \frac{12x}{2x + 1} = 6$$

1. He concludes: “Since the result of the calculation is $6$, the original limit is $6$”.

1. a) Identify and explain in detail the conceptual error Mateo made in his final conclusion. Justify what the value $L = 6$ actually represents by referring to the definition and derivation of the number $e$.
2. b) Write down the correct value of the limit.

Consider analyzing the asymptotic behavior of the following quotient at infinity:

$$L = \lim_{x \to +\infty} \frac{x^{10} + 500 \cdot x^2}{1,02^x}$$

1. a) Estimate: Before applying any algebraic rule, reflect on the numerator and the denominator. The numerator contains a high-degree power ($x^{10}$), while the denominator has an exponential base very close to $1$ ($1,02$). Reason intuitively whether you think the fraction will tend to $0$, to a positive real number, or to $+\infty$ when $x$ reaches values of the order of $x = 10\,000$.
2. b) Check: Formally apply the hierarchy of infinities ($\ln x \ll x^k \ll a^x$ with $a > 1$) to deduce the exact value of $L$. Assess whether your previous estimate was correct and justify why the constant multiplicative rate of an exponential always dominates any polynomial, no matter how large its degree.

A precision sensor in a nanotechnology laboratory measures the attractive force between two particles. The signal intensity $S$ as a function of the distance $x$ (in nanometers) is given by the following function:

$$S(x) = \frac{\sqrt{x+5} - 3}{x-4}$$

The figure shows the graph of $S(x)$: for $x=4$ the function is not defined (point with the empty circle).

Calculate the value the signal approaches as the distance $x$ gets closer to $4$ nanometers. Explain which mathematical technique you used to resolve the indeterminate form.

In the study of news propagation on social networks, it is estimated that the number of users reached $U$ (in thousands) after $t$ hours follows the expression:

$$U(t) = \frac{5t^2 - 3t + 2}{2t^2 + \sqrt{t^4 + 1}}$$

Calculate the limit of users reached as time increases indefinitely ($t \to +\infty$). Does the function have a horizontal asymptote? Justify your answer by comparing the degrees of the dominant terms.

Calculate the following limit by resolving the indeterminate form $\infty - \infty$ using the conjugate expression:

$$\lim_{x \to +\infty} \left( \sqrt{x^2 + 6x} - x \right)$$

Sketch a graph showing what this result means regarding the distance between the radical function and the line $y = x$.

You can do it on paper and send a photo with your answer.

Given the rational function depending on a parameter $k \in \mathbb{R}$:

$$f(x) = \frac{x^3 - kx^2 + 4}{x-2}$$

Find the value of $k$ so that the limit $\lim_{x \to 2} f(x)$ is finite. Once this value is found, calculate the value of the limit.

The growth of a bird population in a protected nature reserve is governed by the following idealised continuous compound growth law:

$$P(t) = \left( \frac{3t + 5}{3t - 2} \right)^{2t}$$

Calculate the limit of the population as time $t$ tends to infinity. Express the result in terms of the number $e$.

Calculate the value of the following limits by comparing the orders of magnitude at infinity of the functions involved (exponentials, powers, and logarithms):

1. a) $\lim_{x \to +\infty} \frac{\ln(x^{10} + 5)}{x}$

1. b) $\lim_{x \to +\infty} \frac{3^x}{x^2 + 100x}$

1. c) $\lim_{x \to +\infty} \frac{e^x + x^5}{2e^x - \sqrt{x}}$

Consider the piecewise function $f(x)$:

$$f(x) = \begin{cases} \frac{x^2 - 9}{x - 3} & \text{if } x < 3 \\ \sqrt{2x + a} & \text{if } x \ge 3 \end{cases}$$

Calculate the value of the parameter $a$ so that the limit of the function exists at $x = 3$. What is the value of this limit?

Calculate the limit at $x = 0$ of the following trigonometric expression, using equivalent infinitesimals or known trigonometric identities:

$$\lim_{x \to 0} \frac{x \cdot \text{tg}(3x)}{\cos(x) - 1}$$

The National Statistics Institute (INE) projects the demographic evolution of a rural region in the province of Zamora affected by depopulation. The number of inhabitants, $P(t)$ (in thousands of people), after $t$ years have elapsed since 2024, is modeled by the function:

$$P(t) = \frac{15t + 8}{\sqrt{4t^2 + 7t + 1} + t}$$

1. a) Calculate the estimated population at the initial instant $t = 0$ and interpret the exact number of inhabitants.
2. b) Calculate $\lim_{t \to +\infty} P(t)$. To resolve the indeterminate form $\frac{\infty}{\infty}$, divide the numerator and denominator explicitly by the highest power of $t$, algebraically justifying how this factor is brought inside the square root.
3. c) Based on the limit found, interpret whether the region tends to become deserted in the long run or whether its population stabilizes at a finite limiting value. Express this value as a specific number of residents.

The following limit is known to exist and to be a finite, non-zero real number:

$$\lim_{x \to 0} \frac{\sqrt{1 + ax} - (1 + 2x)}{b \cdot x^2} = -\frac{3}{2}$$

where $a$ and $b$ are constant real parameters with $b \neq 0$.

1. a) Multiply and divide the quotient by the conjugate expression $\sqrt{1 + ax} + (1 + 2x)$ and expand the numerator, grouping the terms by decreasing powers of $x$.
2. b) Taking into account that the denominator has a quadratic factor $x^2$, reason why it is essential for the linear term in $x$ of the numerator to vanish so that the limit does not diverge to infinity. From this deduction, determine the required value of $a$.
3. c) With the value of $a$ now fixed, simplify the fraction by eliminating the indeterminate form $\frac{0}{0}$ and find the value of $b$ needed for the limit to be exactly $-\frac{3}{2}$.

A multiplayer video game server has detected that the number of active users, $N(t)$, in thousands, follows an exponential growth model as a function of time $t$ (in months) defined by the expression:

$$N(t) = \left( \frac{3t^2 + 5t + 1}{3t^2 + 2t} \right)^{2t}$$

Calculate the limit of active users as time tends to infinity ($t \to +\infty$). Does the player population stabilise at a finite value or grow indefinitely?

When designing a new ramp for a skatepark, engineers need to ensure that the transition between two sections of the track is smooth (continuous). The height of the ramp $h(x)$ in metres is given by the following function:

$$h(x) = \begin{cases} \dfrac{\ln(1 + kx)}{x} & \text{if } x > 0 \\ \cos(x) + a & \text{if } x \le 0 \end{cases}$$

Determine the values of the parameters $k$ and $a$ such that $\displaystyle \lim_{x \to 0} h(x)$ exists and is equal to the initial height of the ramp, which is $2$ metres.

Calculate the value of the following limit using the comparison of infinities or L'Hôpital's Rule, analysing in detail the order of magnitude of each term:

$$\lim_{x \to +\infty} \frac{e^x + x^{10}}{2e^x - \ln(x^5 + 1)}$$

Which term dominates the growth of the function in the numerator and which in the denominator?

A satellite describes a trajectory whose radial distance to the centre of a planet, in thousands of kilometres, is given by $d(x) = \sqrt{x^2 + ax} - \sqrt{x^2 - 1}$, where $x$ represents the mission time. Given that after a long time ($x \to +\infty$) the distance stabilises at $4$ thousand kilometres, determine the value of the orbital constant $a$.

Give reasons to determine the value of the parameters $a$ and $b$ (where $a, b \in \mathbb{R}$) such that the following equality of limits holds:

$$\lim_{x \to 0} \frac{ax^2 + bx + 1 - e^x}{x^2} = 3$$

Study the behaviour of the function $f(x) = \left( \cos x \right)^{1/x^2}$ as $x$ approaches zero. To do so, calculate:

$$\lim_{x \to 0} (\cos x)^{1/x^2}$$

Explain why this limit leads to an indeterminate form of the type $1^\infty$ and solve it using natural logarithms or the specific formula for this type of limit.

Let the function be $g(x) = \dfrac{x^3 - 2x^2 + k}{x^2 - 4}$.

1. a) Calculate the value of $k$ so that the limit of the function as $x \to 2$ is finite.

1. b) Once you have found $k$, calculate that limit.

1. c) Classify the type of discontinuity that the function has at $x = 2$ and at $x = -2$.

Calculate the limit of the following sequence, which frequently appears in continuous compound interest and probability problems:

$$\lim_{n \to \infty} \left( \sqrt{n^2 + n} - n \right)^n$$

Note: This exercise requires careful treatment of the inner indeterminacy before evaluating the overall power.

To calculate the indeterminate limit $\lim_{x \to 0} \frac{\sqrt[3]{1 + 6x} - 1}{\tan(2x)}$, two theoretical approaches are suggested:

• Method A (Cubic conjugate and equivalent infinitesimals): Use the identity $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ to rationalize the numerator and apply the appropriate trigonometric equivalence in the denominator.

• Method B (L'Hôpital's Rule): Verify that the conditions of differentiability and indeterminacy are met, and differentiate the numerator and denominator.

1. a) Fully carry out the solution using Method A. Explain why it is valid to substitute $\tan(2x)$ by its equivalent infinitesimal in this specific case and why it would not be valid if it were inside a subtraction.
2. b) Carry out the solution using Method B, calculating in detail the derivative of the cube root and evaluating the resulting limit.
3. c) Evaluate both methods: which one carries a lower risk of error in exam algebra? Also explain why a change of variables of the form $t = \sqrt[3]{1 + 6x}$ would be significantly more cumbersome due to the presence of the trigonometric function.

In an assessment test, Irene has to solve the limit:

$$\lim_{x \to 0} \frac{x \cdot \sin x - 2(1 - \cos x)}{x^4}$$

Irene writes down the following answer:

“As $x \to 0$, I apply the standard infinitesimals: $\sin x \sim x$ and $(1 - \cos x) \sim \frac{x^2}{2}$. Substituting into the numerator, I get:

$$x \cdot (x) - 2 \cdot \left(\frac{x^2}{2}\right) = x^2 - x^2 = 0$$

Since the numerator becomes zero, the limit is equal to $0$”.

1. a) Explain Irene's conceptual error rigorously. Why can the substitution of equivalent infinitesimals in sums or subtractions lead to a misleading cancellation of the leading terms?
2. b) Prove that the limit is not zero by calculating its exact value through the repeated application of L'Hôpital's Rule (justifying at each step why the hypotheses of the theorem are satisfied).
3. c) Based on the actual result, explain what the first non-zero term of the numerator truly was and why Irene's approximation eliminated the relevant information of the limit.